I don't think immersion is necessarily thinking of the character as yourself. It's more like walking a mile in someone else's shoes. Making the decisions they would make and doing the things they would do. Staying in the role of the character you build. Stopping to ask yourself, would my character do that?
And, to me immersion is just being so into the game I'm playing that if I drop my sword on the ground in game, I may just end up bending over to pick it up off the floor in real life only to realize it's in the game. LOL Immersion is when I loose track of what is happening around me because all of my focus has been given the game. I personally don't imagine I'm in the game but in some ways I am.
I like to become one with my character while knowing that my character is merely a fictional being that I built just like the fictional character I might write about in a book.